Post-Doorway Bounded Bypass in MUTEX Protocols

• Post-doorway bounded bypass is defined as a liveness property where overtaking is bounded only after a process completes a short, bounded doorway phase in the lock routine.
• It refines starvation-freedom by allowing an initial startup period for unfairness while ensuring that subsequent waiting phases have a strict global bound on bypasses.
• Model checking and comparative analysis with protocols like Anderson and Dekker highlight its role in distinguishing different fairness guarantees in MUTEX protocol hierarchies.

Searching arXiv for the specified paper and closely related mutual-exclusion bounded-bypass work. Post-doorway bounded bypass is a liveness property for mutual exclusion protocols that lies strictly between starvation-freedom and bounded bypass. It was introduced to repair a long-used but insufficiently formalized strengthening of starvation-freedom by making precise when a process should count as having begun to wait “in line” for the critical section. In the formulation developed in "On the Notions of Bounded Bypass, and How to Make any Deadlock-Free MUTEX Protocol Satisfy One of Them" (Glabbeek et al., 17 Jun 2026), the key idea is that bypasses are not counted from the very first instruction of lock, but only after completion of a bounded initial segment of the entry protocol, called the doorway. This permits a short startup phase while still imposing a global bound on overtaking once the process has crossed into the genuine waiting phase.

1. Background and motivation

In the MUTEX problem, processes repeatedly execute an entry protocol lock, then a critical section, then an exit protocol unlock, with the fundamental safety requirement that at most one process is ever in the critical section. The paper situates post-doorway bounded bypass within the liveness theory of such protocols. A protocol is deadlock-free if, whenever some process is trying to enter, eventually some process enters a critical section. It is starvation-free if every invocation of lock eventually terminates and lets that same process enter. Bounded bypass is intended as a strengthening of starvation-freedom: a waiting process must not only succeed eventually, but do so after being overtaken only a bounded number of times (Glabbeek et al., 17 Jun 2026).

The motivation for a new notion is twofold. First, the paper argues that Raynal’s definition in terms of “losing at most $f(n)$ competitions” was unsatisfactory because it did not require the process ever to enter its critical section; consequently, a protocol in which every lock is await false would vacuously satisfy that condition while violating both deadlock-freedom and starvation-freedom. Second, older verbal definitions referred to the moment at which a process “requests” or “indicates its desire” to enter, but did not formalize when that occurs. This omission is significant because other processes cannot be expected to defer before such a request becomes observable, and even the first instruction of lock takes time, during which an unbounded number of bypasses may occur (Glabbeek et al., 17 Jun 2026).

This motivates a distinction between a short bounded setup phase and the subsequent waiting phase. The doorway/waiting terminology is attributed in the paper to Lamport. A plausible implication is that post-doorway bounded bypass is designed for protocols whose fairness guarantees begin only after a bounded handshake or initialization sequence, rather than from the first elementary action.

2. Exact definition

The paper first repairs ordinary bounded bypass by defining it as follows: a MUTEX protocol satisfies bounded bypass whenever there is a function $f(n)$ such that every process that invokes the lock enters the critical section, and this happens after losing at most $f(n)$ competitions from the moment in which it completes the execution of the first instruction of the lock (Glabbeek et al., 17 Jun 2026). A process $i$ loses a competition with process $j$ when $j$’s lock overlaps in time with $i$’s and terminates earlier.

The post-doorway variant is then given in the paper’s exact formal definition under the heading “Definition (Doorway bounded bypass)”:

A MUTEX protocol satisfies post-doorway bounded bypass whenever there is a function $f(n)$ and an initial portion of the lock whose execution requires only a bounded number of elementary actions (the doorway) such that every process that invokes the lock enters the critical section, and this happens after losing at most $f(n)$ competitions from the moment in which it completes the doorway. (Glabbeek et al., 17 Jun 2026)

The difference from ordinary bounded bypass is therefore solely the reference point from which overtaking is counted. Ordinary bounded bypass starts the count after completion of the first instruction of lock; post-doorway bounded bypass starts after completion of a bounded initial segment of lock, the doorway. The doorway must be an initial portion of lock and must terminate after only a bounded number of elementary actions (Glabbeek et al., 17 Jun 2026).

Operationally, the property says that a process may be treated unfairly during a short bounded setup period, but once it has crossed the doorway it is in the waiting region and the number of times other processes can overtake it is globally bounded by some function $f(n)$. The bypasses that are ignored are exactly those occurring during execution of the bounded initial code fragment designated as the doorway (Glabbeek et al., 17 Jun 2026).

3. Position in the liveness hierarchy

The paper organizes four liveness notions for mutual exclusion protocols: starvation-freedom, bounded bypass, intermittent bounded bypass, and post-doorway bounded bypass. Starvation-freedom requires that every lock invocation eventually succeeds. Bounded bypass adds a bound on competition losses after the first instruction. Intermittent bounded bypass allows bypasses to be ignored during at most $f(n)$0 designated interrupting write intervals. Post-doorway bounded bypass allows bypasses to be ignored during a bounded initial doorway (Glabbeek et al., 17 Jun 2026).

The implication structure proved in Theorem $f(n)$1 is:

• bounded bypass $f(n)$2 post-doorway bounded bypass,
• bounded bypass $f(n)$3 intermittent bounded bypass,
• post-doorway bounded bypass $f(n)$4 starvation-freedom,
• intermittent bounded bypass $f(n)$5 starvation-freedom,
• starvation-freedom $f(n)$6 deadlock-freedom. (Glabbeek et al., 17 Jun 2026)

The paper further proves that all missing implications fail. In particular, post-doorway bounded bypass is strictly stronger than starvation-freedom, strictly weaker than bounded bypass, and incomparable with intermittent bounded bypass (Glabbeek et al., 17 Jun 2026). This incomparability is explained by the different kinds of exemptions each notion allows. Post-doorway bounded bypass can ignore reads and time between doorway instructions, but only in an initial bounded prefix. Intermittent bounded bypass can ignore selected writes anywhere in the execution, even writes by other processes, but only during those write intervals.

This suggests that post-doorway bounded bypass is not merely a relaxed variant of bounded bypass; it isolates a specific fairness pattern in which initial setup is exempted, while the later waiting phase remains globally regulated.

4. Canonical examples and separations

The paper uses Anderson’s two-process protocol as the primary example showing why post-doorway bounded bypass is needed. It proves that the protocol does not satisfy standard bounded bypass, because one process can execute the first line of lock, then sleep, while the other repeatedly enters the critical section. However, it does satisfy post-doorway bounded bypass if the doorway is taken to include the first four lines. More precisely, Proposition $f(n)$7 states: if $f(n)$8 terminates the execution of line 7 at time $f(n)$9, then $f(n)$0 can bypass $f(n)$1 at most once after time $f(n)$2. Thus, in the atomic-register model, Anderson’s protocol satisfies post-doorway bounded bypass with bound $f(n)$3 for $f(n)$4, while failing standard bounded bypass (Glabbeek et al., 17 Jun 2026).

Model checking refined this example further. If the doorway in Anderson’s protocol is chosen to end after line 2 instead, the protocol still satisfies post-doorway bounded bypass, but with bound $f(n)$5 rather than $f(n)$6 (Glabbeek et al., 17 Jun 2026). This makes the example especially instructive: the property depends not only on the protocol, but also on a justified choice of bounded doorway.

Dekker’s algorithm serves as the canonical separation between starvation-freedom and post-doorway bounded bypass. The paper shows that Dekker is starvation-free under atomic registers, but not bounded bypass, not post-doorway bounded bypass, and not intermittent bounded bypass. The crucial observation is that the only admissible doorway is the first line of lock, because the subsequent loop can execute an unbounded number of times before completion. Hence, for Dekker, post-doorway bounded bypass collapses to ordinary bounded bypass; since bounded bypass fails, so does post-doorway bounded bypass (Glabbeek et al., 17 Jun 2026).

The paper also uses the Bar-David transformation as a separation from the intermittent notion. In safe and regular registers, the transformed protocol satisfies intermittent bounded bypass but not bounded bypass, and the paper indicates that it does not satisfy post-doorway bounded bypass either. This is one half of the argument that intermittent bounded bypass and post-doorway bounded bypass are unrelated (Glabbeek et al., 17 Jun 2026).

5. Models, transformations, and quantitative bounds

A major result of the paper concerns an algorithm proposed by Yoah Bar-David in 1998 for enhancing the liveness properties of any deadlock-free MUTEX protocol. In the setting of atomic registers, the paper proves that this transformation upgrades any deadlock-free mutual exclusion protocol to a bounded bypass one, with a tight bound

$f(n)$7

and therefore also to a post-doorway bounded bypass protocol as an immediate consequence. In the setting of safe and regular registers, the same transformation ensures only intermittent bounded bypass, with

$f(n)$8

and not bounded bypass (Glabbeek et al., 17 Jun 2026).

The paper summarizes the status of the main examples as follows.

Protocol / model	Bounded bypass	Post-doorway bounded bypass
Dekker / atomic	no	no
Anderson / atomic	no	$f(n)$9
Bar-David / atomic	$i$0	yes
Bar-David / safe/regular	no	no

The same summary notes that Bar-David / safe/regular satisfies intermittent bounded bypass with $i$1 and $i$2 (Glabbeek et al., 17 Jun 2026).

These results clarify the status of post-doorway bounded bypass as an analytical target. It is not the main guarantee proved for the Bar-David transformation: in atomic registers a stronger property is obtained directly, while in safe and regular registers the guaranteed target is intermittent bounded bypass, not post-doorway bounded bypass. Post-doorway bounded bypass therefore functions both as a definitional repair and as a middle property that captures protocols, such as Anderson’s, whose fairness begins after a bounded startup segment.

6. Logical formalization and model checking

The paper formalizes bounded bypass and post-doorway bounded bypass in modal $i$3-calculus for model checking. For a set of processes $i$4, the model includes actions $i$5 for starting lock, $i$6 for finishing lock, and $i$7 for starting and finishing a write. For post-doorway bounded bypass, the model introduces a special action $i$8 marking when process $i$9 leaves the doorway; in the model this is implemented “as an extra tag on the final action of the doorway” (Glabbeek et al., 17 Jun 2026).

The intended meaning of the formula $j$0 is that, each time $j$1 starts its lock, once it performs the distinguished action marking the end of the doorway, at most $j$2 bypasses by other processes may happen before $j$3 finishes its own lock. The paper states that the formula for ordinary bounded bypass is exactly the same, except that $j$4 is fixed to coincide with the end of the first instruction of $j$5’s lock, making bounded bypass a special case of post-doorway bounded bypass (Glabbeek et al., 17 Jun 2026).

The formalization is only partial: the formula does not itself quantify over all doorway placements or verify that the chosen doorway requires only boundedly many elementary actions. Those aspects must be checked externally by reasoning about the chosen initial segment and by trying reasonable doorway placements (Glabbeek et al., 17 Jun 2026).

Model checking confirmed the principal claims. For Dekker, with the doorway at the end of the first line, no finite bound checked up to 100 suffices. For Anderson, a doorway after the fourth line gives bound $j$6, while a doorway after the second line gives bound $j$7. For the Bar-David algorithm in atomic registers, model checking confirmed the tight bounded-bypass bound $j$8, improving an earlier non-tight $j$9 bound. The paper also reports that model checking helped uncover ambiguities in the characterization of interrupting assignments for intermittent bounded bypass and had a positive impact on the theoretical development by identifying and correcting small mistakes and ambiguities in definitions and proofs (Glabbeek et al., 17 Jun 2026).

Post-doorway bounded bypass therefore occupies a distinct place in mutual-exclusion liveness theory. It is weaker than full bounded bypass because it exempts a bounded initial prefix of lock, but stronger than starvation-freedom because, once the doorway is crossed, overtaking is globally bounded. In that sense, it provides a precise fairness notion for protocols whose meaningful waiting phase begins only after a bounded startup segment (Glabbeek et al., 17 Jun 2026).